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Mathematics is the study of numbers, quantity, space, pattern, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Used for calculation, it is considered the most important subject. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

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A homotopy from a circle around a sphere down to a single point.
Image credit: Richard Morris

The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).

The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from a c-sphere into a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology. (Full article...)

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A line integral is an integral where the function to be integrated, be it a scalar field as here or a vector field, is evaluated along a curve. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). A detailed explanation of the animation is available. The key insight is that line integrals may be reduced to simpler definite integrals. (See also a similar animation illustrating a line integral of a vector field.) Many formulas in elementary physics (for example, W = F · s to find the work done by a constant force F in moving an object through a displacement s) have line integral versions that work for non-constant quantities (for example, W = ∫C F · ds to find the work done in moving an object along a curve C within a continuously varying gravitational or electric field F). A higher-dimensional analog of a line integral is a surface integral, where the (double) integral is taken over a two-dimensional surface instead of along a one-dimensional curve. Surface integrals can also be thought of as generalizations of multiple integrals. All of these can be seen as special cases of integrating a differential form, a viewpoint which allows multivariable calculus to be done independently of the choice of coordinate system. While the elementary notions upon which integration is based date back centuries before Newton and Leibniz independently invented calculus, line and surface integrals were formalized in the 18th and 19th centuries as the subject was placed on a rigorous mathematical foundation. The modern notion of differential forms, used extensively in differential geometry and quantum physics, was pioneered by Élie Cartan in the late 19th century.

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